Existence of optimal controls for systems governed by parabolic partial differential equations with Cauchy boundary conditions

Summary This paper considers the existence of optimal controls for systems governed by a second order parabolic partial differential equation in divergence form with Cauchy conditions. As preliminary results, theorems concerning the convergence of the sequence of weak solutions corresponding to a sequence of admissible controls are proved. Two general forms of criteria are considered. The first one is taken as a function of the weak solution of the system, and the other is taken as a function of the solution of the system and control. Several theorems and corollaries on the existence of optimal controls are then presented.